The basic tenet of the present work is the assumption of the lack of external and fixed time in the Universe. This assumption is best embodied by general relativity, which replaces the fixed space-time structure with the gravitational field, which is subject to dynamics. The lack of time does not imply the lack of evolution but rather brings to the forefront the role of internal clocks which are some largely arbitrary internal degrees of freedom with respect to which the evolution of timeless systems can be described. We take this idea seriously and try to understand what it implies for quantum mechanics when the fixed external time is replaced by an arbitrary internal clock. We put the issue in a solid, mathematically rigorous framework. We find that the dynamical interpretation of a quantum state of a timeless system depends on the employed internal clock. In particular, we find that the continuous spectra of well-known dynamical observables like the position of a free particle on the real line may turn discrete if measured in unusual clocks. We discuss the meaning of our result for attempts at quantization of global gravitational degrees of freedom.
The basic tenet of the present work is the assumption of the lack of external and fixed time in the Universe. This assumption is best embodied by general relativity, which replaces the fixed space-time structure with the gravitational field, which is subject to dynamics. The lack of time does not imply the lack of evolution but rather brings to the forefront the role of internal clocks which are some largely arbitrary internal degrees of freedom with respect to which the evolution of timeless systems can be described. We take this idea seriously and try to understand what it implies for quantum mechanics when the fixed external time is replaced by an arbitrary internal clock. We put the issue in a solid, mathematically rigorous framework. We find that the dynamical interpretation of a quantum state of a timeless system depends on the employed internal clock. In particular, we find that the continuous spectra of well-known dynamical observables like the position of a free particle on the real line may turn discrete if measured in unusual clocks. We discuss the meaning of our result for attempts at quantization of global gravitational degrees of freedom.
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
This project explores a fully timeless formulation of quantum theory motivated by the Page–Wootters mechanism and canonical quantum gravity. The central assumption is that the universe is described by a single global quantum state ∣Ψ⟩ satisfying a Wheeler–DeWitt–type Hamiltonian constraint, with time appearing only as an internal, relational variable defined by “clock” subsystems. I formulate a simple multi–clock framework, introduce a clock neutrality conjecture, and test it in an explicit finite-dimensional toy model. The model consists of two three-tick internal clocks and a qubit system, with joint step relations that encode a static global state over a 3×3 grid of clock readings. Imposing multi-clock consistency selects a constrained family of states in which the two clock evolutions commute on the support of ∣Ψ⟩. Within the decoherent histories framework, I then show that for histories expressible with either clock, the probabilities for relational events agree whenever the two clocks implement the same underlying physical evolution. In this sense the toy model realises a discrete version of clock neutrality: decoherent-histories probabilities are independent of which internal clock is used as the parameter. The construction does not yet yield new empirical predictions, but it isolates a precise structural condition that any timeless, deterministic description of the universe would need to satisfy in order to be compatible with clock-independent low-energy physics.
Abstract The operational approach to time is a cornerstone of relativistic theories, as evidenced by the notion of proper time. In standard quantum mechanics, however, time is an external parameter. Recently, many attempts have been made to extend the notion of proper time to quantum mechanics within a relational framework. Here, we use similar ideas combined with the relativistic mass-energy equivalence to study an accelerating massive quantum particle with an internal clock system. We show that the ensuing evolution from the perspective of the particle’s internal clock is non-Hermitian. This result does not rely on specific implementations of the clock. As a particular consequence, we prove that the effective Hamiltonian of two gravitationally interacting particles is non-Hermitian from the perspective of the clock of either particle.
Quantum timeless approaches solve the problem of time by recovering the usual unitary evolution of quantum theory relative to a clock in a stationary quantum universe. For some Hamiltonians of the universe, such as those including an interaction term with the clock, the dynamics is substantially altered and can be nonunitary. This work derives necessary and sufficient conditions for the relative dynamics to be unitary and finds the general form of the unitary evolution operator. A physical interpretation of these conditions is given in terms of the clock’s rate. Unitary dynamics is associated with rates that are constant in time and independent of the clock’s internal structure.
Time in relativity theory has a status different from that adopted by standard quantum mechanics, where time is considered as a parameter measured with reference to an external absolute Newtonian frame. This status strongly restricts its role in the dynamics of systems and hinders any formulation to merge quantum mechanics with general relativity, specifically when considering quantum gravity. To overcome those limitations, several authors tried to construct an operator which is conjugate to the Hamiltonian of quantum systems implementing some essential features of the relativistic time. These formulations use the concept of internal or intrinsic time instead of the universal coordinate time used in textbooks. Furthermore, recently it is remarked that the consideration of time with relativistic features could enhance the analysis techniques in quantum information processing and have an impact on its status in causal orders and causal structures of quantum information. The role of clocks, their accuracy and stability has become an important issue in quantum information processing. This article present a substantiative review of recent works which reflect the possibility of utilizing quantum time, measured by quantum clock devised according to Page-Wootters scheme, to stand as a resource for quantum information processing.
We study classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the physical time, which is assumed to be discrete. In this way, a physical clock with discrete states is introduced, which presently is still treated as decoupled from the system. This is motivated by the recent discussion of ``timeless'' reparametrization invariant models, where discrete physical time has been constructed based on quasi-local observables. Employing the path-integral formulation of classical mechanics developed by Gozzi et al., we show that these deterministic classical systems can be naturally described as unitary quantum mechanical models. We derive the emergent quantum Hamiltonian in terms of the underlying classical one. Such Hamiltonians typically need a regularization - here performed by discretization - in order to arrive at models with a stable groundstate in the continuum limit. This is demonstrated in several examples, recovering and generalizing a model advanced by 't Hooft.